Problem: Find the value of $n$ that satisfies $\frac{1}{n+1} + \frac{2}{n+1} + \frac{n}{n+1} = 3$.
Answer: Combining the fractions on the left gives $\dfrac{n+3}{n+1} = 3$.  Multiplying both sides by $n+1$ gives $n+3 = 3(n+1)$.  Expanding the right side gives $n+3 = 3n+3$.  Subtracting $n$ and 3 from both sides gives $0=2n$, so $n=\boxed{0}$.